After looking into this, I have to say that the definition of "monergism" seems to depend on who you ask. However, the Council of Trent explicitly rejected any form of monergism with the following:
If anyone says that the sinner is justified by faith alone in the sense that nothing else is required by way of cooperation in order to obtain the grace of justification and that it is not at all necessary that he should be prepared and disposed by the movement of his will, let him be anathema.
All Protestants, as far as I can tell, reject this synergism. (At least going back to Arminius. More on that later.)
Again as far as I can tell, there are three primary Protestant positions on matters of Salvation as relating to election and monergism: Lutheranism, Calvinism and Arminianism. Understanding the differences is challenging, but a commonality seems to be that humans, of their own capacity, are incapable of cooperating in Salvation. While Catholics affirm that the process must be started by God, Catholicism seems to teach that active participation becomes a requirement at some point, which Protestants would argue violates their principle of Sola Gratia ("grace alone").
(If I seem to be using a lot of equivocations, it is because individual believers will differ in their understanding. As a result, it can be difficult to pin down official doctrines, and individuals within a body may not always agree with each other. Many Protestants would claim that, while the Roman Catholic church is in error as an organization, there are nevertheless true believers within their ranks... just as there are heretics within the ranks of all branches of Protestantism.)
By contrast, monergism asserts that the individual can do nothing of their own will to assist in Salvation. Differences arise, however, in whether an individual can freely work against Salvation. Irresistible Grace, in particular, says "no".
Ironically, if I understand correctly, Arminians are actually more synergistic than Catholics, asserting that one "makes a choice for Christ". Some sects might view this as Salvation-by-works, perhaps even "worse" (in their view) than what Roman Catholics teach. That said, whether this is consistent with Arminius' own teachings is debatable. (More generally, "Arminianism" probably needs to be more finely divided; however, as will be further explained, we aren't actually focused on Arminians for this Question.)
All of these beliefs lie on a spectrum, where one end indicates that Salvation is entirely in God's hands, and the other indicates that it is entirely in Man's hands. On the whole, Christianity tends toward the former, while man-made religions tend toward the latter. However, it is likely that some man-made religions involve the assistance of a deity, and some "Christian" sects (particularly the Church of Jesus Christ of Latter Day Saints, which is widely regarded as heretical and non-Christian by "mainstream" Christians) are further from the "God only" end than others.
I would summarize the differences between sects thusly:
Branch |
Beliefs |
Calvinists |
Believe that Salvation is entirely in God's hands and that Man can do nothing to either obtain nor reject Salvation. Grace is "irresistible". |
Lutherans |
Believe that Salvation is entirely an act of God, but that Man can resist. |
Catholics |
Believe that Salvation is initiated by God but that the full process requires cooperation on the part of the believer. |
Arminians |
Believe that one must "make a choice for Christ" in order to be Saved. |
This article also uses TULIP to discuss the differences between Lutherans, Calvinists and Arminians.
This Question, therefore, seems to be asking about Lutheranism specifically. Thus, the best answer I can give is that Lutherans believe in monergism in the sense that Man is incapable of contributing to his own Salvation in any active way, but is capable of detracting from (and potentially even losing) Salvation. By contrast, Calvinists believe Salvation cannot be lost, while Catholics believe Man must actively cooperate.
To use an analogy, Calvinists see humans as cars that are broken; the car has no say in whether the mechanic (God) fixes them or leaves them broken, and no ability to interfere with being fixed. Lutherans and Catholics, however, see humans as something like Frankenstein's monster; dead (spiritually) to start with, but brought to life by God. However, Lutherans would say that all we can "do" is to let God continue fixing us, while Catholics would say that at some point the believer has to actively help.
To be fair, this is a subtle and difficult-to-grasp distinction, and particularly as it lies in between Calvinism and Catholicism, it's understandable why the difference is confusing.
Further reading: